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To find \( x \) in the given triangle, we can use trigonometric relationships and properties of special right triangles. The triangle provided has a \( 60^\circ \) angle, a \( 45^\circ \) angle, and a \( 90^\circ \) angle, which implies it is divided into two special right triangles:

1. A \( 45^\circ - 45^\circ - 90^\circ \) triangle
2. A \( 30^\circ - 60^\circ - 90^\circ \) triangle

First, let's analyze the given side \( 7\sqrt{3} \) which is opposite to the \( 60^\circ \) angle. This makes it the side opposite the \( 60^\circ \) angle in the \( 30^\circ - 60^\circ - 90^\circ \) triangle.

In a \( 30^\circ - 60^\circ - 90^\circ \) triangle:
- The side opposite the \( 30^\circ \) angle is \( \frac{1}{2} \) of the hypotenuse.
- The side opposite the \( 60^\circ \) angle is \( \frac{\sqrt{3}}{2} \) of the hypotenuse.

Given \( 7\sqrt{3} \) is the side opposite the \( 60^\circ \) angle, we can find the hypotenuse \( h \):

\[
7\sqrt{3} = \frac{\sqrt{3}}{2} h
\]

Solving for \( h \):

\[
h = \frac{7\sqrt{3} \times 2}{\sqrt{3}} = 14
\]

Now, the side opposite the \( 30^\circ \) angle (which is also the adjacent side to the \( 60^\circ \) angle) is half of the hypotenuse:

\[
\frac{14}{2} = 7
\]

Next, we consider the \( 45^\circ - 45^\circ - 90^\circ \) triangle. In such a triangle, the sides are in the ratio \( 1 : 1 : \sqrt{2} \). The hypotenuse is \( x \), and each leg is \( 7 \):

\[
x = 7\sqrt{2}
\]

So, the missing length \( x \) is \( 7\sqrt{2} \).

However, the options given do not have \( 7\sqrt{2} \) directly but have another format. Let's look at the choices:
- \( A = \frac{21\sqrt{2}}{2} \)
- \( B = 7 \)

Converting \( 7\sqrt{2} \) to the given format:

\[
7\sqrt{2} = \frac{14\sqrt{2}}{2}
\]

None of the options fit \( 7\sqrt{2} \). It seems there is a mistake. 

Would you like further assistance or have any questions?

Here are some related questions for further exploration:
1. What are the properties of a \( 30^\circ - 60^\circ - 90^\circ \) triangle?
2. How do you find the hypotenuse in a \( 45^\circ - 45^\circ - 90^\circ \) triangle?
3. What is the significance of special right triangles in trigonometry?
4. How can trigonometric ratios be used to solve for unknown sides in right triangles?
5. What are some applications of trigonometry in real-life problems?
6. How do you verify the accuracy of your solution in trigonometry problems?
7. Can you derive the side ratios for special right triangles?
8. What are the common mistakes to avoid when solving trigonometry problems?

**Tip:** Always verify your final answer by checking if it fits all the given conditions and constraints in the problem.

Learn how to find the missing side length x in a triangle using trigonometric relationships and properties of special right triangles. Detailed step-by-step solution included. Suitable for high school students studying trigonometry and geometry.

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